1. Field of the Invention
The present invention relates to an apparatus and a method for evaluating a radial vibration of a rotating body and a rotation unit with the rotating body evaluated by the radial vibration evaluation method, and an apparatus and a method for evaluating a rotational accuracy of a rolling bearing and the rolling bearing evaluated by the rotational accuracy evaluation method, in which the radial vibration asynchronous with the rotational speed of the rotating body for example non repeatable round out of such as a bearing or a spindle with a bearing incorporated therein is evaluated on the basis of frequency analysis. In particular, the present invention relates to the evaluation of a maximum (minimum) amplitude value and a maximum (minimum) azimuth of a specific frequency component of which the amplitude varies depending on a radial azimuth.
2. Description of the Related Art
A rotating body of such as rolling bearing or a spindle with a rolling bearing incorporated therein generates radial vibration due to the circularity of a bearing portion, or the like. Therefore, the rolling bearing and spindle including such a rotating body may be a vibration source generating serious vibration of a structure such as a machine tool including them.
Such a rotating body generates radial vibration also at the time of constant-speed rotation. The radial vibration contains radial vibration components asynchronous with the rotational speed of the rotating body. The asynchronous radial vibration components are called xe2x80x9cNRRO (Non Repeatable Round Out) vibration componentsxe2x80x9d. The NRRO vibration component is constituted by a plurality of frequencies. The frequencies of the NRRO vibration component are determined in accordance with predetermined calculation expressions on the basis of the geometric sizes of inner and outer rings, and rolling bodies such as balls in a bearing incorporated in a rotation unit, the accuracy of form thereof and the constant-speed rotational speed of the rotating body. For the NRRO vibration components constituted by the plurality of frequencies, it is known that there is the NRRO vibration component constituted by a part of frequencies wherein the magnitude (amplitude) of vibration changes depending on the azimuth in the direction of rotation of the rotating body (hereinafter called xe2x80x9cdepending on the azimuthxe2x80x9d).
On the other hand, particularly in a hard disk device, vibration of a disk due to vibration of a ball bearing used in a rotational shaft of the hard disk becomes a main cause of error in positioning a magnetic head. Therefore, the ball bearing needs to have strict rotational accuracy.
For the provision of such a ball bearing satisfying the rotational accuracy required for the hard disk device, it is necessary to evaluate the NRRO vibration components quantitatively to thereby remove a ball bearing having NRRO vibration components unsuitable for positioning the magnetic head. It is further necessary to mainly evaluate the NRRO vibration components constituted by frequencies (hereinafter referred to as xe2x80x9cspecific frequenciesxe2x80x9d) near to the resonance frequency of the hard disk device particularly selectively to thereby remove a ball bearing which may otherwise cause resonance of the hard disk device. Here, xe2x80x9cspecific frequenciesxe2x80x9d is referred in xe2x80x9cVibration and Noisexe2x80x9d that is described in PP. 919-963, Chapter 25, in xe2x80x9cRolling Bearing Analysis (Third Edition) xe2x80x9d written by T. A. Harris and published by John Wiley and sons, inc. (1991), for example, fc, fci, Zfc, Zfci, fR defined by expressions (25.14)-(25.18) in pp. 950-951. The xe2x80x9cspecific frequencyxe2x80x9d is also described in REFERENCES 25.4 and 25.10 in pp. 962-963 of the same, that is, 25.4) O. Gustafsson, T. Tallian et al., xe2x80x9cFinal Report on the Study of Vibration Characteristics of Bearingsxe2x80x9d, U.S. Navy Contract NObs-78552, U.S. Navy Index No. NE071 200 (Dec. 6, 1963) and 25.10) O. Custaisson and U. Rimrott, xe2x80x9cMeasurement of Surface Waviness of Rolling-Element Bearing Partsxe2x80x9d, SAE Paper 195C (June 1960).
Generally, the NRRO vibration components of the specific frequency includes an NRRO vibration component dependent on the azimuth and an NRRO vibration component independent of the azimuth. Both NRRO vibration components are respectively constituted by a plurality of frequencies. That is, the specific frequency of the NRRO vibration component dependent on the azimuth has a plurality of frequencies, and the specific frequency of the NRRO vibration component independent of the azimuth has a plurality of frequencies. Accordingly, for example, if a NRRO vibration component of the specific frequency depends on the azimuth, the evaluation of such NRRO vibration component is carried out in such manner that magnitude (amplitude) of the NRRO vibration component constituted by the plurality of frequencies dependent on the azimuth is measured in accordance with each azimuth over all the azimuths in the direction of rotation of the rotating body; and a maximum amplitude value and an azimuth exhibiting the maximum amplitude value are determined among the magnitudes measured for NRRO vibration component constituted by the plurality of frequencies.
As one of methods for evaluating the radial vibration of the rotating body at all azimuths in the direction of rotation of the rotating body, there is known a method in which displacement sensors such as displacement measuring units for measuring the radial vibration are disposed in two places near the outer circumference of the rotating body so that the azimuths of the displacement sensors are different from each other, and radial vibration at a third azimuth different from the two azimuths is evaluated by use of vibration values measured at the two azimuths. In the method for evaluating the radial vibration, two displacement sensors are disposed so that the radial vibration may be measured in two directions (x- and y-directions) perpendicular to a rotational shaft (or rotating body) of the rotation unit and perpendicular to each other (FIG. 12). When t is the time for measuring vibration and x(t) and y(t) are measured x- and y-direction vibration components respectively, radial vibration f(t, xcex8) at an azimuth xcex8 wherein xcex8 is an angle rotated in the direction of rotation of the rotating body from the x axis on which one of the displacement sensors is disposed is given by the following expression:
f(t, xcex8)=x(t)cos xcex8+y(t)sin xcex8
The amplitude of the NRRO vibration component is generally evaluated by the maximum value of the fluctuation width in the case where the NRRO vibration components are extracted every rotating period of the rotating body and superposed on each other. Here, a NRRO evaluation value means the maximum amplitude selected from the amplitudes of the NRRO vibration components at respective azimuths, and the maximum azimuth xcex8max means an azimuth exhibiting the NRRO evaluation value.
Generally, the radial vibration f(t, xcex8) includes other vibration components than the NRRO vibration component dependent on the azimuth. For this reason, the amplitude of the NRRO vibration component of the specific frequency dependent on the azimuth must be selectively obtained from the radial vibration f(t, xcex8) by frequency analysis in order to evaluate the amplitude of the NRRO vibration component of the specific frequency dependent on the azimuth.
Generally, frequency analysis using Fourier transform is performed to selectively obtain the amplitude of the specific frequency from vibration constituted by the plurality of frequencies. In the aforementioned method of evaluating the radial vibration, in the case where radial vibration f (t, xcex8max) expressing the NRRO evaluation value and the maximum azimuth xcex8max are obtained, even if the amplitude of the NRRO vibration component of the specific frequency dependent on the azimuth is to be evaluated by use of frequency spectra {Fk(xcex8max), k=0, 1, . . . Nxe2x88x921} obtained by Fourier transform of a sequence of N discrete values {fn(xcex8max)=f(nxcex94t, xcex8max), n=0, 1, . . . Nxe2x88x921} obtained by sampling the radial vibration f(t, xcex8max) by xcex94t, frequency spectra corresponding to the specific frequency in the frequency spectral distribution obtained by Fourier transform of the maximum azimuth xcex8max may not be always the maximum value in all azimuths.
This is because as described above, the specific frequency is generally constituted by a plurality of frequencies, and the maximum azimuth xcex8max is an azimuth maximizing the synthetic amplitude determined by the relative relations among frequency, amplitude and phase in each the NRRO component of specific frequency constituted by the plurality of frequencies dependent on the azimuth; however, the NRRO vibration component of each frequency generally exhibits maximum amplitude at azimuths different from one another.
For this reason, the true maximum azimuth xcex8max in each the NRRO vibration component of the specific frequency need to be obtained as follows: a frequency spectral distribution is obtained from radial vibration f(t, xcex8) {0xe2x89xa6xcex8 less than 2xcfx80} at each azimuth by frequency analysis using Fourier transform; a frequency spectra corresponding to the specific frequency are obtained at each azimuth; the vibration levels of the frequency spectra are compared with one another all over the azimuths in accordance with each frequency, so that a frequency spectrum exhibiting the maximum vibration level is selected from the frequency spectra in accordance with each frequency; a frequency spectrum exhibiting the maximum vibration level is further selected from the selected frequency spectra for respective frequencies; and an azimuth exhibiting the selected frequency spectrum is obtained.
In the frequency analysis using Fourier transform at each azimuth, however, FFT (Fast Fourier Transform) operation needs to be repeated at each azimuth xcex8 if the azimuth xcex8 is taken finely to increase accuracy in calculation of the true maximum azimuth xcex8max. Hence, it is necessary to perform a great deal of calculation. There is a problem that NRRO vibration components of radial vibration of the rotating body in a production line cannot be evaluated in real time.
Further, when the radial vibration of the rolling bearing is observed at one point on a fixed ring, vibration caused by form errors in a rotating ring and rolling elements is observed equally in any position on the fixed ring. It is however known that vibration caused by form error in the fixed ring is observed with different magnitude in accordance with a location of measurement. Such vibration cannot be evaluated correctly unless a maximum amplitude is found all over the circumference of the fixed ring. Therefore, there can be conceived of a method for finding a maximum value of a specific frequency component by repeated frequency analysis while displacing a sensor for measuring vibration in the circumferential direction relative to the bearing or spindle. However, such a mechanism is not easy, and it takes much time for measurement.
Next, for example, when an inner ring rotates if the raceway of an outer ring which is a fixed ring has very small waviness, there is generated vibration having one specific frequency corresponding to a pair of numbers (for example NZ+1 and NZxe2x88x921; where N is integer, Z is number of rolling elements) in such polygonal waviness. On the other hand, when the outer ring rotates, there is generated vibration having a pair of frequencies corresponding to the pair of numbers in such raceway waviness. At the same time, there is generated vibration having one specific frequency in accordance with a pair of numbers in the raceway waviness of the inner ring which is a fixed ring. That is, there is a case where both or one of vibration having one specific frequency and vibration having a pair of frequencies must be selected strictly as a harmful vibration component in accordance with the operating condition of the rolling bearing. To this end, a mechanism which can observe both the rotating conditions of the inner ring and the outer ring may be provided. However, such a mechanism increases both the complexity and the price of a test apparatus.
It is a first object of the present invention to provide an apparatus and a method for evaluating a radial vibration of a rotating body, in which NRRO vibration components of the radial vibration of the rotating body in a production line can be evaluated in real time without any great deal of calculation, and a rotation unit with the rotating body evaluated by the radial vibration evaluation method.
It is a second object of the present invention to provide an apparatus and a method for evaluating a rotational accuracy of a rolling bearing, in which vibration values corresponding to respective numbers in raceway waviness of a fixed ring are obtained from vibration of one specific frequency caused by a pair of numbers, and the rolling bearing evaluated by the rotational accuracy evaluation method. In order to attain the foregoing object, vibration of the bearing is observed simultaneously from two radial directions which are not opposed (that is, not at 180xc2x0). Thus, there is provided a method for calculating the vibration values from solutions of simultaneous equations using, as coefficients, vibration values of a specific frequency caused by the fixed ring obtained on the basis of frequency analysis results of the observed vibration.
It is a third object of the present invention to provide a method in which a maximum (minimum) value of vibration of a specific frequency caused by a pair of numbers in the fixed ring and a maximum (minimum) azimuth exhibiting the maximum (minimum) value of vibration are calculated on the basis of a pair of vibration values owing to the fixed ring obtained as described above.
It is a forth object of the present invention to provide a method in which a maximum (minimum) value of vibration synthesized integrally when a rotating ring is used as a fixed ring is estimated and evaluated from measurement of vibration components of a pair of frequencies caused by a pair of numbers appearing in the rotating ring.
Further, since the fixed ring is not rigid, vibration accompanied with elastic deformation of the fixed ring caused by the passage of the rolling elements is observed at a point of measurement on the circumference of the fixed ring. The frequency of the passage of the rolling elements is equal to a specific frequency caused by the form error of the fixed ring. Thus, there is a fear that a vibration component caused by the form error of the fixed ring cannot be evaluated correctly.
It is therefore a fifth object of the present invention to provide a method in which even in such a condition, the influence of the component caused by the passage of the rolling elements is eliminated by additional observation of vibration in a different, third direction, so that the vibration component of the fixed ring can be evaluated correctly.
It is a sixth object of the present invention to provide a bearing marked in a position where the vibration becomes maximal, or a position where the vibration becomes minimal which is distant by 90xc2x0 from the maximum vibration position, so that a spindle or a motor the vibration of which is thus the lowest in a specific direction can be produced.
To achieve the first object of the present invention, there is provided with a method of evaluating a radial vibration of a rotating body with the amplitude changing depending on an azimuth in a direction of rotation of the rotating body. The method comprises:
measuring the radial vibration from two mutually different directions at a specific frequency asynchronous with the rotational speed of the rotating body;
transforming vibration components of the radial vibration obtained by measurement from the two directions into frequency spectra respectively by Fourier transform;
calculating the amplitude of the radial vibration in accordance with each azimuth in the direction of rotation of the rotating body on the basis of the transformed frequency spectra obtained in the two directions; and
selecting and evaluating a maximum amplitude of the radial vibration and an azimuth exhibiting the maximum amplitude on the basis of calculated amplitudes of the radial vibration for azimuths.
In the method of evaluating the radial vibration of the rotating body according to the present invention, vibration components obtained by measurement from two directions are transformed into frequency spectra respectively by Fourier transform, so that the amplitude of radial vibration at the specific frequency is calculated in accordance with each azimuth on the basis of the transformed frequency spectrum obtained in the two directions. Hence, there is no necessity of performing Fourier transform in accordance with each azimuth, so that NRRO vibration components of the radial vibration of the rotating body in a production line can be evaluated in real time without any great deal of calculation.
An amplitude calculation process used in the present invention for obtaining the amplitude of the radial vibration of the rotating body at the specific frequency in accordance with an azimuth xcex8 will be described below.
First, vibration components x(t) and y(t) in x and y directions are measured. A sequence of discrete values {xn=x(nxcex94t), yn=y(nxcex94t), n=0, 1, . . . Nxe2x88x921} sampled at N points by dividing the vibration components x(t) and y(t) by At respectively are subjected to Fourier transform. The Fourier transform in this process is discrete Fourier transform based on FFT operation. Hence, when discrete points on a frequency axis are replaced by k, frequency spectra {Xk, Yk, k=0, 1, . . . Nxe2x88x921} (hereinafter referred to as xe2x80x9cfrequency sample valuesxe2x80x9d) expressed by Xk and Yk are obtained.
Then, frequency spectra Xh, Yh corresponding to a frequency point h corresponding to a specific frequency are selected from the frequency spectra {Xk, Yk, k=0, 1, . . . Nxe2x88x921}. A square of a frequency sample value of radial vibration f(t, xcex8) at an azimuth xcex8 in accordance with the frequency point h is calculated on the basis of the selected frequency spectra Xh, Yh.
First, as described above, the radiation vibration f(t, xcex8) at the azimuth xcex8 is given by the following expression (1).
f(t,xcex8)=x(t)cos xcex8+y(t)sin xcex8xe2x80x83xe2x80x83(1) 
When both sides of this expression (1) are subjected to Fourier transform, the square |Fh(xcex8)|2 of the frequency sample value of the radial vibration f(t, xcex8) at the azimuth xcex8 in accordance with the frequency point h is given by the following expression (2).                                                                                           "LeftBracketingBar"                                                            F                      h                                        ⁡                                          (                      θ                      )                                                        "RightBracketingBar"                                2                            =                                                "LeftBracketingBar"                                                                                    X                        h                                            ⁢                      cos                      ⁢                                              xe2x80x83                                            ⁢                      θ                                        +                                                                  Y                        h                                            ⁢                      sin                      ⁢                                              xe2x80x83                                            ⁢                      θ                                                        "RightBracketingBar"                                2                                                                                        =                                                (                                                                                    X                        h                                            ⁢                      cos                      ⁢                                              xe2x80x83                                            ⁢                      θ                                        +                                                                  Y                        h                                            ⁢                      sin                      ⁢                                              xe2x80x83                                            ⁢                      θ                                                        )                                ⁢                                  (                                                                                    X                        h                        *                                            ⁢                      cos                      ⁢                                              xe2x80x83                                            ⁢                      θ                                        +                                                                  Y                        h                        *                                            ⁢                      sin                      ⁢                                              xe2x80x83                                            ⁢                      θ                                                        )                                                                                                        =                                                                                          "LeftBracketingBar"                                              X                        h                                            "RightBracketingBar"                                        2                                    ⁢                                      cos                    2                                    ⁢                  θ                                +                                                                            "LeftBracketingBar"                                              Y                        h                                            "RightBracketingBar"                                        2                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                  θ                                +                                                      (                                                                                            X                          h                                                ⁢                                                  Y                          h                          *                                                                    +                                                                        Y                          h                                                ⁢                                                  X                          h                          *                                                                                      )                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                  θcos                  ⁢                                      xe2x80x83                                    ⁢                  θ                                                                                                        =                                                                                          "LeftBracketingBar"                                              X                        h                                            "RightBracketingBar"                                        2                                    ⁢                                      cos                    2                                    ⁢                  θ                                +                                                                            "LeftBracketingBar"                                              Y                        h                                            "RightBracketingBar"                                        2                                    ⁢                                      sin                    2                                    ⁢                  θ                                +                                  2                  ⁢                                      Re                    ⁡                                          (                                                                        X                          h                                                ⁢                                                  Y                          h                          *                                                                    )                                                        ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                  θcos                  ⁢                                      xe2x80x83                                    ⁢                  θ                                                                                        (        2        )            
In the expression (2), |A| is the absolute value of a complex number, Xh* and Yh* are conjugate to complex numbers Xh and Yh respectively, and Re (B) is a real part of a complex number B.
On this occasion, if the sequence of discrete values of x(t) and y(t) obtained at N points is a sequence of real values, the sequence of discrete values {f(nxcex94t, xcex8), n =0, 1, . . . Nxe2x88x921} of the radial vibration f(t, xcex8) at the azimuth xcex8 is also a sequence of real values. Hence, a RMS (Room Mean Square) amplitude value of the frequency sample value of the radial vibration f(t, xcex8) at the azimuth xcex8 at the frequency point h is 2xc2xd|Fh(xcex8). On the basis of the expression (2), the RMS amplitude value of the frequency sample value of the radial vibration f(t, xcex8) at the azimuth 0 in accordance with the frequency point h is given by the following expression (3).                                                                                           2                                      1                    /                    2                                                  |                                                      F                    h                                    ⁡                                      (                    θ                    )                                                  |                            =                              xe2x80x83                            ⁢                                                2                                      1                    /                    2                                                  ⁢                                  {                                      |                                          X                      h                                        ⁢                                          |                      2                                        ⁢                                                                                                                        cos                            2                                                    ⁢                          θ                                                +                                            |                                              Y                        h                                                              ⁢                                          |                      2                                        ⁢                                                                                            sin                          2                                                ⁢                        θ                                            +                                                                                                                                                                                xe2x80x83                                ⁢                                  2                  ⁢                                      Re                    (                                                                  X                        h                                            ⁢                                              Y                        h                        *                                                              )                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                  θcosθ                                }                                            1                /                2                                                                        (        3        )            
The RMS amplitude value of the frequency sample value of the radial vibration f(t, xcex8) at the azimuth xcex8 in accordance with the frequency point h is calculated as the amplitude of the radial vibration of the rotating body at the azimuth xcex8 at the frequency point h, and then this process is terminated.
On this occasion, the maximum RMS amplitude value selected from the RMS amplitude values of the frequency sample values of the radial vibration f(t, xcex8) calculated for frequency points h at all azimuths in directions of rotation of the rotating body is used as an RMS evaluation value in evaluation of NRRO vibration components, and the azimuth xcex8 exhibiting the RMS evaluation value is used as the maximum azimuth xcex8 max.
Preferably, the amplitude of the radial vibration may be calculated on the basis of the frequency spectra obtained in the two directions at the specific frequency at each azimuth obtained by subdividing the azimuths of from 0 to xcfx80 in the direction of rotation of the rotating body, so that the maximum amplitude may be selected from the amplitude values calculated at the respective azimuths.
When, for example, the RMS amplitude value of the frequency sample value of the radial vibration f(t, 2jxcfx80/M) at the azimuth 2jxcfx80/M in the case where the direction of rotation of the rotating body at the frequency point h is divided into M parts is calculated as the amplitude of the radial vibration of the rotating body at the specific frequency at the azimuth xcex8, the RMS amplitude value of the frequency sample value of the radial vibration f (t, 2jxcfx80/M) at azimuth 2jxcfx80/M at the frequency point h is given by the following expression (4).                                                                                           2                                      1                    /                    2                                                  |                                                      F                    h                                    ⁡                                      (                                          2                      ⁢                      j                      ⁢                                              xe2x80x83                                            ⁢                                              π                        /                        M                                                              )                                                  |                            =                              xe2x80x83                            ⁢                                                2                                      1                    /                    2                                                  ⁢                                  {                                      |                                          X                      h                                        ⁢                                          |                      2                                        ⁢                                                                                            cos                          2                                                ⁡                                                  (                                                      2                            ⁢                            j                            ⁢                                                          xe2x80x83                                                        ⁢                                                          π                              /                              M                                                                                )                                                                    +                                                                                                                                                              xe2x80x83                            ⁢                              |                                  Y                  h                                ⁢                                  |                  2                                ⁢                                                                            sin                      2                                        ⁡                                          (                                              2                        ⁢                        j                        ⁢                                                  xe2x80x83                                                ⁢                                                  π                          /                          M                                                                    )                                                        +                                      2                    ⁢                                          Re                      (                                                                        X                          h                                                ⁢                                                  Y                          h                          *                                                                    )                                                                                                                                                                                xe2x80x83                                ⁢                                                                            sin                      ⁡                                              (                                                  2                          ⁢                          j                          ⁢                                                      xe2x80x83                                                    ⁢                                                      π                            /                            M                                                                          )                                                              ⁢                                          cos                      ⁡                                              (                                                  2                          ⁢                          j                          ⁢                                                      xe2x80x83                                                    ⁢                                                      π                            /                            M                                                                          )                                                                              }                                                            1                /                2                                                                        (        4        )            
In the expression (4), the absolute values |Fh(2jxcfx80/M) of frequency sample values at azimuths 2jxcfx80/M =0 to T are equal to the absolute values |Fh(2jxcfx80/M+xcfx80)| of frequency sample values at azimuths 2(j+M/2)xcfx80/M=2jxcfx80/M+xcfx80=xcfx80 to 2xcfx80. Hence, the RMS amplitude values of frequency sample values of the radial vibration f(t, 2jxcfx80/M) at azimuths 2jxcfx80/M can be calculated on the basis of the azimuths 2jxcfx80/M=0 to xcfx80. As a result, the RMS amplitude values of frequency sample values of radial vibration f(t, 2jxcfx80/M) at azimuths 2jxcfx80/M need not be calculated over the whole circumference in the direction of rotation of the rotating body. Hence, the time required for calculation can be shortened.
On this occasion, the maximum RMS amplitude value selected from the RMS amplitude values of frequency sample values of the radial vibration f (t, 2jxcfx80/M) at the frequency point h calculated at azimuths 2jxcfx80/M=0 to n in the direction of rotation of the rotating body is used as an RMS evaluation value, and an azimuth 2jxcfx80/M exhibiting the maximum RMS evaluation value is used as the maximum azimuth xcex8max.
Preferably, the amplitude of the radial vibration may be calculated on the basis of the frequency spectra obtained in the two directions at the specific frequency at each azimuth obtained by subdividing the azimuths of from 0 to xcfx80/2 in the direction of rotation of the rotating body, so that the maximum amplitude may be selected from the amplitude values calculated at the respective azimuths.
When, for example, cos(2jxcfx80/M) and sin(2jxcfx80/M) in the expression (4) are replaced by xe2x88x92sin(2jxcfx80/M) and cos(2jxcfx80/M) respectively, the RMS amplitude value at an azimuth (2jxcfx80/M+xcfx80/2) larger by xcfx80/2 than the azimuth 2jxcfx80/M is given by the following expression (5).                                           2                          1              /              2                                |                                    F              h                        ⁡                          (                                                2                  ⁢                  j                  ⁢                                      xe2x80x83                                    ⁢                                      π                    /                    M                                                  +                                  π                  /                  2                                            )                                |                =                              2                          1              /              2                                ⁢                      {                          |                              X                h                            ⁢                              |                2                            ⁢                                                                    sin                    2                                    ⁡                                      (                                          2                      ⁢                      j                      ⁢                                              xe2x80x83                                            ⁢                                              π                        /                        M                                                              )                                                  +                                  "AutoLeftMatch"                                      |                                          Y                      h                                        ⁢                                          |                      2                                        ⁢                                                                                            cos                          2                                                ⁡                                                  (                                                      2                            ⁢                            j                            ⁢                                                          xe2x80x83                                                        ⁢                                                          π                              /                              M                                                                                )                                                                    -                                              2                        ⁢                                                  Re                          (                                                                                    X                              h                                                        ⁢                                                          Y                              h                              *                                                                                )                                                ⁢                                                  sin                          ⁡                                                      (                                                          2                              ⁢                              j                              ⁢                                                              xe2x80x83                                                            ⁢                                                              π                                /                                M                                                                                      )                                                                          ⁢                                                  cos                          ⁡                                                      (                                                          2                              ⁢                              j                              ⁢                                                              xe2x80x83                                                            ⁢                                                              π                                /                                M                                                                                      )                                                                          ⁢                                                  }                                                      1                            /                            2                                                                                                                                                                                                      (        5        )            
Accordingly, when the expressions (4) and (5) are used, the RMS amplitude values of frequency sample values of the radial vibration f(t, 2jxcfx80/M) at the frequency point h at the azimuths 2jxcfx80/M =0 to n can be calculated by a simple operation of calculating sin (2jxcfx80/M) and cos (2jxcfx80/M) at the azimuths 2jxcfx80/M=0 to xcfx80/2. Hence, the RMS amplitude values of frequency sample values of the radial vibration f (t, 2jxcfx80/M) at the azimuths 2jxcfx80/M need not be calculated over the half circumference in the direction of rotation of the rotating body. Hence, the time required for calculation can be shortened more.
Preferably, the maximum amplitude of radial vibration and the azimuth exhibiting the maximum amplitude may be deduced on the basis of the frequency spectra in the two directions and the addition theorem of trigonometric functions.
A maximum amplitude deducing process used in the present invention for deducing the maximum amplitude of the radial vibration and the azimuth exhibiting the maximum amplitude on the basis of the frequency spectra in the two directions and the addition theorem of trigonometric functions will be described below.
In the maximum amplitude deducing process in the present invention, the expression (2) is expressed by the following expression (6) using ah=(|Xh|2+|Yh|2)/2, bh=(|Xh|2xe2x88x92|Yh|2)/2 and ch=(XhYh*+YhXh*)/2=2Re(XhYh*).                                                                                           "LeftBracketingBar"                                                            F                      h                                        ⁡                                          (                      θ                      )                                                        "RightBracketingBar"                                2                            =                                                a                  h                                +                                                      b                    h                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  θ                                +                                                      c                    h                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                  2                  ⁢                  θ                                                                                                        =                                                a                  h                                +                                                                            (                                                                        b                          h                          2                                                +                                                  c                          h                          2                                                                    )                                                              1                      /                      2                                                        ⁢                                      cos                    ⁡                                          (                                                                        2                          ⁢                          θ                                                -                                                  φ                          h                                                                    )                                                                                                                              (        6        )            
In the expression (6), "PHgr"h is equal to tanxe2x88x921 (ch/bh). Accordingly, from the expression (6), the maximum azimuth Oman for maximizing the square |Fh(xcex8)|2 of the frequency sample value is given by the following expression (7).                                                                         θ                max                            =                                                φ                  h                                /                2                                                                                        =                                                                    [                                                                  tan                                                  -                          1                                                                    ⁡                                              [                                                  2                          ⁢                                                                                    Re                              ⁡                                                              (                                                                                                      X                                    h                                                                    ⁢                                                                      Y                                    h                                    *                                                                                                  )                                                                                      /                                                          (                                                                                                                                    "LeftBracketingBar"                                                                          X                                      h                                                                        "RightBracketingBar"                                                                    2                                                                -                                                                                                      "LeftBracketingBar"                                                                          Y                                      h                                                                        "RightBracketingBar"                                                                    2                                                                                            )                                                                                                      ]                                                              ]                                    /                  2                                ⁢                                  xe2x80x83                                ⁢                and                                                                                                        θ                max                            =                                                                    φ                    h                                    /                  2                                ±                π                                                                        (        7        )            
Further, the maximum value of the square |Fh(xcex8)|2 of the frequency sample value is given by the following expression (8).                                                                                           "LeftBracketingBar"                                                            F                      h                                        ⁡                                          (                                                                        φ                          h                                                /                        2                                            )                                                        "RightBracketingBar"                                2                            =                              xe2x80x83                            ⁢                                                a                  h                                +                                                      (                                                                  b                        h                        2                                            +                                              c                        h                        2                                                              )                                                        1                    /                    2                                                                                                                          =                              xe2x80x83                            ⁢                                                                    (                                                                                            "LeftBracketingBar"                                                      X                            h                                                    "RightBracketingBar"                                                2                                            +                                                                        "LeftBracketingBar"                                                      Y                            h                                                    "RightBracketingBar"                                                2                                                              )                                    /                  2                                +                                                                                                        xe2x80x83                            ⁢                                                                    {                                                                                            (                                                                                                                    "LeftBracketingBar"                                                                  X                                  h                                                                "RightBracketingBar"                                                            2                                                        -                                                                                          "LeftBracketingBar"                                                                  Y                                  h                                                                "RightBracketingBar"                                                            2                                                                                )                                                2                                            +                                                                        (                                                                                                                    X                                h                                                            ⁢                                                              Y                                h                                *                                                                                      +                                                                                          Y                                h                                                            ⁢                                                              X                                h                                *                                                                                                              )                                                2                                                              }                                                        1                    /                    2                                                  /                2                                                                                        =                              xe2x80x83                            ⁢                                                (                                                                                    "LeftBracketingBar"                                                  X                          h                                                "RightBracketingBar"                                            2                                        +                                                                  "LeftBracketingBar"                                                  Y                          h                                                "RightBracketingBar"                                            2                                        +                                          "LeftBracketingBar"                                                                        X                          h                          2                                                +                                                  Y                          h                          2                                                                    "RightBracketingBar"                                                        )                                /                2                                                                        (        8        )            
On this occasion, the maximum value of 21/2 Fh (0) is the RMS evaluation value and is given by the following expression (9).
2xc2xd|Fh(xcfx86h/2)|=(|Xh|2+|Yh|2+Xh2+Yh2|)xc2xdxe2x80x83xe2x80x83(9) 
Accordingly, when the expressions (7) and (9) are used, the maximum azimuth ƒmax and the RMS evaluation value can be calculated by a simple operation of calculating |Xh|2, |Yh|2, Re(XhYh*) and |Xh2+Yh2| on the basis of Xh and Yh whenever the RMS evaluation value is to be calculated. Hence, the RMS amplitude values at respective azimuths in the direction of rotation of the rotating body need not be calculated. Hence, the time required for calculation can be shortened extremely.
Preferably, a judgment as to the rotating performance of the rotating body is made on the basis of comparison between the maximum amplitude and a predetermined value. When, for example, the maximum amplitude exceeds a predetermined value, a decision may be made that the rotating performance of the rotating body is bad.
The judgment as to the rotating performance can be made easily because a decision is made that the rotating performance is bad when the maximum amplitude exceeds a predetermined value.
Incidentally, the use of the complex values Xh and Yh may be replaced by the use of the amplitude and phase thereof. In the method of evaluating the radial vibration of the rotating body on this occasion, it is easy to deduce expressions equivalent to the expressions (1) to (9). Hence, the description of the equivalent expressions will be omitted.
On the basis of evaluation according to the radial vibration evaluation method as described above, a fixed member of a rotation unit, such as a fixed ring of a rolling bearing, a housing of a spindle may be marked in a position of the maximum azimuth exhibiting the maximum RMS evaluation value or minimum azimuth being distant by 90xc2x0 from the maximum azimuth.
To achieve the second to sixth objects of the present invention, there is provided with a method for evaluating a rotational accuracy of a rolling bearing comprising:
measuring the radial vibration of a fixed ring of the rolling bearing by use of two vibration measuring sensors disposed circumferentially with a phase xcex1 to thereby obtain sensor signals;
making the sensor signals discrete through an A/D converter to thereby obtain two pieces of synchronizing digital data;
Fourier-transforming the digital data to thereby obtain vibration values F(m) and Fxcex1(m) of an order m of angular velocity Zxcfx89c;
obtaining unknown quantities Amzxe2x88x921exe2x88x92jxcex8 and Amz+1ejxcex8 by use of the vibration values F(m) and Fxcex1(m) on the basis of the following expressions (11) and (12):
Amzxe2x88x921exe2x88x92jxcex8={ejxcex1F(m)}/2j sin xcex1xe2x80x83xe2x80x83(11) 
Amz+1ejxcex8={Fxcex1(m)xe2x88x92exe2x88x92jxcex1F(m)}/2j sin xcex1xe2x80x83xe2x80x83(12) 
(wherein m designates an order of vibration, Z designates the number of rolling elements, j designates an imaginary number such that j2=1, xcfx89cdesignates an angular velocity of revolution of the rolling elements, and xcex8 designates a center angle between an unknown reference position on the fixed ring and one of the vibration measuring sensors);
obtaining, from the unknown quantities, RMS values of vibration components caused by mZxe2x88x921 (th) polygon and mZ+1(th) polygon respectively in accordance with the following expressions (13) and (14):
RMS value of component of mZxe2x88x921(th) polygon=2|Amzxe2x88x921|=2|Amzxe2x88x921exe2x88x92jxcex8|xe2x80x83xe2x80x83(13) 
RMS value of component of mZ+1(th) polygon=2|Amz+1|=2|Amz+1ejxcex8|xe2x80x83xe2x80x83(14); 
xe2x80x83and
evaluating the rotational accuracy of the rolling bearing on the basis of the RMS values.
The two vibration measuring sensors are disposed at two points on the fixed ring of the rolling bearing in order to observe the radial vibration. Signals from the two vibration measuring sensors are taken in a computer through the A/D converter so as to be used as two synchronizing sequences of sampled values. Vibration values (RMS values) corresponding to a pair of numbers in the shape of the fixed ring are obtained from solutions of complex simultaneous linear equations with two unknowns, using as coefficients, complex values of a desired frequency component selected from Fourier transforms of the respective sequences, and a function of an angle between the positions where the vibration measuring sensors are disposed.
The sum of absolute values of these two vibration values (or the difference therebetween) is set to a maximum (minimum) value of vibration depending on the shape of the fixed ring.
On the other hand, from the complex values of the desired frequency component selected from the respective Fourier transforms, and the angle between the positions where the two sensors are disposed, the azimuth in which vibration becomes maximal (minimal) is calculated as a relative angle with respect to each of the two sensors. Since the maximum vibration values and the azimuths exhibiting the maximum vibration values are known thus, the rotational accuracy of the rolling bearing can be evaluated by comparing the maximum vibration values and their azimuths with threshold values.
Incidentally, a maximum amplitude value and a minimum amplitude value of angular velocity mZxcfx89c are expressed by the following expressions (15) and (16):
maximum RIMS value=2(|Amzxe2x88x921|+|Amz+1|)xe2x80x83xe2x80x83(15) 
minimum RMS value=2∥Amzxe2x88x921|xe2x88x92|Amz+1∥xe2x80x83xe2x80x83(16) 
Further, phases of the maximum amplitude value and the minimum amplitude value of the angular velocity mZxcfx89c are:
maximal at xcex3o and xcex3o+xcfx80 and minimal at xcex3oxc2x1xcfx80/2 if |xcex3o|xe2x89xa6xcfx80/4, and minimal at xcex3o and xcex3o+xcfx80 and maximal at xcex3oxc2x1xcfx80/2 if xcfx80/4 less than |xcex3o|xe2x89xa6xcfx80/2, respectively when |F(m)|2 cos 2xcex1+|Fxcex1(m)|2xe2x88x92{F (m) F*xcex1(m)+F* (m) Fxcex1(m)}cos xcex1xe2x89xa60; and
minimal at xcex3o and xcex3o+xcfx80 and maximal at xcex3oxc2x1xcfx80/2 if |xcex3o|xe2x89xa6xcfx80/4, and maximal at xcex3o and xcex3o+xcfx80 and minimal at xcex3o+xcfx80/2 if xcfx80/4 less than |xcex3o|xe2x89xa6xcfx80/2, respectively, when |F(m)|2cos 2xcex1+|Fxcex1(m)|2xe2x88x92{F(m)F*xcex1(m)+F*(m)Fxcex1(m)}cos xcex1 greater than 0 (providing 2xcex3o is given by expression (46) as described below). Incidentally, F* (m) used in the specification (including the scope of claim for a patent) is assumed to be conjugate to F(m).
Further, in accordance with expression (49) as described below, by use of the vibration values of the fixed ring, RMS values of vibration components caused by mZxe2x88x921 (th) polygon and m mZ+1 (th) polygon respectively when a rotating ring and the fixed ring are used reversely are set as:
RMS value of component of mZxe2x88x921(th) polygon=2|Bmzxe2x88x921|xe2x80x83xe2x80x83(17) 
RMS value of component of mZ+1(th) polygon=2|Bmz+1|xe2x80x83xe2x80x83(18); 
and a maximum amplitude value and a minimum amplitude value of angular velocity mZxcfx89c are set as:
maximum RMS value=2(|Bmzxe2x88x921|+|Bmz+1|)xe2x80x83xe2x80x83(19) 
minimum RMS value=2∥Bmzxe2x88x921|xe2x88x92|Bmz+1∥xe2x80x83xe2x80x83(20) 
That is, the sum of the absolute values (or the difference therebetween) of the pair of vibration values corresponding to a pair of numbers of pairs in the rotating ring selected from at least one of the above-mentioned Fourier transforms provides a maximum (minimum) value of vibration of the above-mentioned one specific frequency when the rotating ring is regarded as a fixed ring.
Further, another vibration measuring sensor is provided at a phase xcex2, and in accordance with expressions (50) to (54) as described below, RMS values of vibration components caused by mZxe2x88x921(th) polygon and mZ+1(th) polygon respectively are set as:
RMS value of component of mZxe2x88x921(th) polygon=2|Amz+1|xe2x80x83xe2x80x83(21) 
RMS value of component of mZ+1(th) polygon=2|Amz+1|xe2x80x83xe2x80x83(22); 
and a maximum amplitude value and a minimum amplitude value of angular velocity mZxcfx89c are set as:
maximum RMS value=2(Amzxe2x88x921|+|Amz+1|)xe2x80x83xe2x80x83(23) 
minimum RMS value=2∥Amzxe2x88x921|xe2x88x92|Amz+1∥xe2x80x83xe2x80x83(24) 
That is, when the third vibration measuring sensor is disposed, in the same manner as and at the same time that the signals from the two vibration measuring sensors, a signal from the third vibration measuring sensor is taken in a computer through the A/D converter, so that the three signals are used as three synchronizing sequences of sampled values. Vibration values corresponding to a pair of numbers of pairs in the shape of the fixed ring and an vibration value corresponding to the elastic deformation of the fixed ring accompanied with the passage of the rolling elements are obtained from solutions of complex simultaneous linear equations with three unknowns, using, as coefficients, complex values of a desired frequency component selected from Fourier transforms of the three sequences, and a function of angles among the three vibration measuring sensors. The fixed ring is thus evaluated in the same manner as described above.
In a similar manner, when F(m)=Fxe2x80x2(m)xe2x88x92Dmzexe2x88x92jmzxcex8 and Fxcex1 (m) F=xcex1xe2x80x2(m)xe2x88x92Dmzexe2x88x92jmz(xcex8+xcex1) are set, phases of the maximum amplitude value and the minimum amplitude value of the angular velocity mZxcfx89c are:
maximal at xcex3o and xcex3o+xcfx80 and minimal at xcex3o+xcfx80/2 if |xcex3o|xe2x89xa6xcfx80/4, and minimal at xcex3o and xcex3o+xcfx80 and maximal at xcex3o+xcfx80/2 if xcfx80/4 less than |xcex3o less than xcfx80/2, respectively, when IF(m) 2cos2a +IFa(n)2 {F(m)F*xcex1(m)+F*(m)Fxcex1(m)}cos xcex1xe2x89xa60; and
minimal at xcex3o and xcex3o+xcfx80 and maximal at xcex3oxc2x1/2 if |xcex3o|xe2x89xa6xcfx80/4, and maximal at xcex3o and xcex3o+xcfx80 and minimal at xcex3oxc2x1xcfx80/2 if xcfx80/4 less than |xcex3o|xe2x89xa6xcfx80/2, respectively, when |F(m)|2 cos 2xcex1+|Fxcex1(m)|2xe2x88x92{F(m)F*xcex1(m)+F*(m)Fxcex1(m)} cos xcex1 greater than 0 (providing 2xcex3o is given by expression (46) as described below).
On the basis of evaluation according to the rotational accuracy evaluation method as described above, the fixed ring is marked in a position where an vibration component is maximal or minimal in the rolling bearing. Accordingly, when such a rolling bearing is installed, the direction of the bearing in which vibration is minimal is aligned with the direction of cutting with a cutting tool or grinding wheel in a processing machine. In a hard disk unit, the direction of the bearing in which vibration is minimal is aligned with the direction of movement of a head. Thus, the influence of vibration caused by the rolling bearing can be reduced.